2013/10/14 by Shigeo Koshitani, Koshitani, Shigeo, Caroline Lassueur +1
Mathematics · #20C20 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT #msc:20C20
paper · pdf · doi:10.48550/arxiv.1310.3647
17 pages. Changes from (v1): S. Koshitani was added as an author. This is an improvement of (v1) focusing on the group of endotrivial modules for finite groups with Klein-four Sylow 2-subgroups. The last section was removed
arxiv created 2014/10/09 · arxiv updated 2014/10/10
We study the finitely generated abelian group T(G) of endo-trivial kG-modules where kG is the group algebra of a finite group G over a field of characteristic p>0. When the representation type of the group algebra is not wild, the group structure of T(G) is known for the cases where a Sylow p-subgroup P of G is cyclic, semi-dihedral and generalized quaternion. We investigate T(G), and more accurately, its torsion subgroup TT(G) for the case where P is a Klein-four group. More precisely, we give a necessary and sufficient condition in terms of the centralizers of involutions under which TT(G) = f-1(X(NG(P))) holds, where f-1(X(NG(P))) denotes the abelian group consisting of the kG-Green correspondents of the one-dimensional kNG(P)-modules. We show that the lift to characteristic zero of any indecomposable module in TT(G) affords an irreducible ordinary character. Furthermore, we show that the property of a module in f-1(X(NG(P))) of being endo-trivial is not intrinsic to the module itself but is decided at the level of the block to which it belongs.